A385182 Values of u in the quartets (1,u,v,w); i.e., values of u for solutions to (1+u) = v*(v+w), in positive integers, with v>1, sorted by nondecreasing values of u; see Comments.

5, 7, 9, 11, 11, 13, 14, 15, 17, 17, 19, 19, 20, 21, 23, 23, 23, 25, 26, 27, 27, 29, 29, 29, 31, 31, 32, 33, 34, 35, 35, 35, 37, 38, 39, 39, 39, 41, 41, 41, 43, 43, 44, 44, 45, 47, 47, 47, 47, 49, 49, 50, 51, 51, 53, 53, 53, 54, 55, 55, 55, 56, 57, 59, 59

Offset: 1

Author

Clark Kimberling, Jun 23 2025

Keywords

• nonn

Comments

A 4-tuple (m,u,v,w) is a quartet if m,u,v,w are positive integers such that m

If m is a prime, then (u,v,w) = (m+2,m+1,m-1) is the first solution (in the defined ordering of triples).

u >= 1 appears A056924(u+1)-1 times. - Pontus von Brömssen, Jul 06 2025

Examples

			First 30 quartets (1,u,v,w):
  m    u    v    w
  1    5    2    1
  1    7    2    2
  1    9    2    3
  1   11    2    4
  1   11    3    1
  1   13    2    5
  1   14    3    2
  1   15    2    6
  1   17    2    7
  1   17    3    3
  1   19    2    8
  1   19    4    1
  1   20    3    4
  1   21    2    9
  1   23    2   10
  1   23    3    5
  1   23    4    2
  1   25    2   11
  1   26    3    6
  1   27    2   12
  1   27    4    3
  1   29    2   13
  1   29    3    7
  1   29    5    1
  1   31    2   14
  1   31    4    4
  1   32    3    8
  1   33    2   15
  1   34    5    2
  1   35    2   16
1*(1+23) = 2*(2+10) = 3*(3+5) = 4*(4+2), so three of the rows are (1,23,2,10), (1,23,3,5), and (1,23,4,2).
		

Crossrefs

Guide to related sequences:

m | u | v | w

--+---------+---------+--------

1 | A385182 | A385183 | A385184

2 | A385592 | A385593 | A385594

3 | A385595 | A385596 | A385597

4 | A385598 | A385599 | A385600

--+---------+---------+--------

Cf. A056924.

Programs

Mathematica

Clear[solnsM];
solnsM[m_, max_] := Module[{ans = {}, rhs = {}, u, v, w, lhs, matching},
Do[Do[AppendTo[rhs, {v*(v + w), v, w}], {w, max}], {v, m*(m + max)}];
rhs = GatherBy[rhs, First];
Do[lhs = m*(m + u); matching = Select[rhs, #[[1, 1]] == lhs &];
If[Length[matching] > 0, Do[AppendTo[ans,
Map[{m, u, #[[2]], #[[3]]} &, matching[[1]]]], {i,
Length[matching]}]], {u, max}];
ans = Flatten[ans, 1];
Select[Union[Map[Sort[{#, RotateLeft[#, 2]}][[1]] &,
Sort[Select[DeleteDuplicates[
ans], {#[[1]], #[[2]]} =!= {#[[3]], #[[4]]} &]]]], #[[1]] == m &]];
TableForm[solns = solnsM[1, 140], TableHeadings -> {None, {"m", "u", "v", "w"}}]
aa = Flatten[solns]
Map[#[[2]] &, solns]    (* u, A385182 *)
Map[#[[3]] &, solns]    (* v, A385183 *)
Map[#[[4]] &, solns]    (* w, A385184 *)
(*Peter J.C.Moses, Jun 15 2025*)

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